Method for Estimating Missing Well Log Data

ABSTRACT

The invention relates generally to the field of oil and gas exploration and specifically to the use of well logs for exploration. This invention is directed to a method for estimating data that would have been collected in a region of a well log where there is a gap. This method uses identified elements in one data set to identify elements in another data set with data values indicative of the same geological characteristic as those in the first data set.

FIELD OF INVENTION

The invention relates generally to the field of oil and gas explorationand specifically to the use of well logs for exploration. This inventionis directed to a method for estimating data that would have beencollected in a region of a well log where there is a gap. This methoduses identified elements in one data set to identify elements in anotherdata set with data values indicative of the same geologicalcharacteristic as those in the first data set.

BACKGROUND OF THE INVENTION

A “well log” comprises data collected along the path of a hole in theground. Such holes are referred to as “wells” in the art of oil and gasexploration. A well log “curve” is a sequential collection of onecategory of data, such as resistivity or gamma ray activity. A well mayhave several logs, and a log may have several curves.

A primary task in the search for oil and gas is to gain an understandingof the distribution and nature of rocks and fluids in the subsurface.This understanding is important for the success and the efficiency ofthe search. Well logs provide direct information about what is in thesubsurface. Data collected by logging wells can have significanteconomic consequences because wells may cost millions of dollars and oildeposits can be worth billions of dollars.

Several logs are commonly acquired from each well. Data are usuallyacquired by lowering sensing tools into the hole by cable. The cableholds the tools and maintains electrical connection with recordingequipment at the surface. Data are acquired by the sensors and “logged”(recorded) at the surface as the tool is pulled up the hole. Data mayalso be acquired by instrumentation at the bottom of the hole whiledrilling is in progress. Data collected in or descriptive of the rockand fluid surrounding the hole fall into the category of “well logdata”.

Well logs provide detailed and direct measurements of rock and fluidproperties in the subsurface. Examples of such measurements are (a)gamma ray intensity, which relates to the types of minerals present; (b)electrical resistance, which relates to the quantity and types offluids; and (c) sonic velocity (the time required for sound to travelfrom sender to receiver), which relates to both rock and fluidproperties. These three examples are illustrative of the hundreds ofwell logs that may be collected.

Unfortunately, gaps are sometimes present in well logs. Well log gapsresult in less information on which to base a model and more uncertaintyregarding what will be encountered when the next well is drilled. Gapsare present for a variety of reasons. Tools may fail or malfunction, oroperators may turn off recording equipment at the wrong time. It may bediscovered after the fact that the wrong interval was logged.

SUMMARY OF THE INVENTION

One embodiment of the invention disclosed herein is directed to a methodfor estimating the data that would have been collected in the region ofa gap in a well log. Another embodiment of the invention disclosedherein is directed to a method for estimating the accuracy andreliability of the rock properties in the gap.

DESCRIPTION OF THE FIGURES

FIG. 1 is a block diagram of a preferred method of the inventiondisclosed herein.

FIG. 2 is a graphical representation of recorded well log data suitablefor use with preferred embodiments of the invention disclosed herein.

FIG. 3 displays well log values at corresponding depths in numericalformat.

FIG. 4 illustrates how similar log values may be identified when onlyone log curve is available.

FIG. 5 illustrates how similar log values may be identified when two logcurves are available.

FIG. 6 illustrates how similar log values may be identified when threelog curves are available.

FIG. 7 illustrates how the principle components method can reduce thenumber of variables.

FIG. 8 illustrates a typical sigmoid function.

FIG. 9 illustrates how density functions are combined to measureconfidence.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Consider the case of a well that has N logs with one log, designated“m”, which has a gap in interval Z, as shown in FIG. 2. Let ˜M refer tologs other than m and let ˜Z refer to the intervals above and below Zover which all logs have values, as shown in FIG. 2. Depths where logsare absent are not used by a preferred method of the invention describedherein. A preferred embodiment of the invention disclosed herein derivesa prediction algorithm that uses a plurality of logs ˜M to predict m inthe intervals ˜Z, and then uses this algorithm to predict values of m inthe interval Z, thus filling the gap.

For notation purposes, the interval Z contains sample locations i, wherevalues v_(i) are to be predicted. Values in other logs, logs in the set˜M, at the same depth as location i collectively constitute the elementp_(i) in the set P. The interval ˜Z contains samples at locations j,where values v_(j) are known. Other data at locations j in the interval˜Z are combined with v_(j) to form the element q_(j). All such elementsin ˜Z are collectively called the set Q.

In a preferred embodiment, the invention disclosed herein is directed toa method for selecting samples of geological data. In a preferredembodiment, the first step of this method is acquiring a data setcomprising multiple data values indicative of at least one geologicalcharacteristic at locations in a subsurface interval of rock, as shownin Block 10 of FIG. 1.

In a preferred embodiment, the second step of this method is identifyinglocations i where (i ∈ {1,n}) in a subsurface interval of rock wherevalues v_(i) are to be predicted, as shown in Block 12 of FIG. 1.

In a preferred embodiment, the third step of this method is identifyingdata values associated with each location i such that each data valueassociated with a location i is an element p_(i) and all elements p_(i)form set P, as shown in Block 14 of FIG. 1.

In a preferred embodiment, the fourth step of this method is identifyinglocations j where (j ∈ {1,m}) in a subsurface interval of rock wheredata values v_(j) from the data set are known, as shown in Block 16 ofFIG. 1.

In another preferred embodiment, locations i and j are depth locations.In another preferred embodiment, i comprises locations in a first welllog curve from a first well which is missing sample values. In anotherpreferred embodiment, p_(i) comprises sample values from a second welllog curve in the first well and at the same depths i. In anotherpreferred embodiment, j are locations in a first well log curve from afirst well which is missing sample values comprising data in wells otherthan the first well at depths where value v_(j) is known.

In a preferred embodiment, the fifth step of this method is definingelements q_(j) to be samples v_(j) and data at the same location assamples v_(j) at a multiplicity of locations j to form set Q, as shownin Block 18 of FIG. 1.

In another preferred embodiment, values v_(j) are known well log values.In another preferred embodiment, the data at the same location assamples v_(j) at a multiplicity of locations j are log values at thesame depth as v_(j).

In a preferred embodiment, the sixth step of this method is for anelement p_(i) in set P, identifying one or more elements q_(j) in set Qwith data values indicative of the same geological characteristic as thep_(i) sample is indicative of, such that the identified elements q_(j)constitutes data elements r_(i), as shown in Block 20 of FIG. 1.

In a preferred embodiment, the seventh step of this method is defining adata set R, comprising data elements r_(i), as shown in Block 22 of FIG.1.

Another preferred embodiment of the invention comprises the firstthrough seventh steps described above plus the eighth step ofassociating an element q_(j) with an element p_(i) by the Euclideann-dimensional distance of q_(j) from p_(i) using only elements q_(j) inset Q identified in step f of claim 1, as shown in Block 24 of FIG. 1.

In a preferred embodiment, the content of elements q_(j) are comparedwith the contents of a given element p_(i). Those elements q_(j) whichare designated “similar” to p_(i) are designated r_(i). The elementsr_(i) are collectively referred to as the set R. For each element p_(i),then, there is a set R which contains elements of similar content. R isthe basis of predicting the value v_(i). If the data are such that R isempty, then v_(i) cannot be predicted.

FIG. 2 displays data from gamma ray, electrical, and sonic logs. Thesethree logs extend from a depth of 3000 meters to 3500 meters. The soniclog shown in FIG. 2 has a gap between 3200 meters and 3250 meters. In apreferred embodiment, sample values v_(i) are to be predicted atlocations i in this depth range. In the ranges of 3043 meters to 3200meters and 3250 meters to 3439 meters, all logs shown in FIG. 2 havevalues. Samples in these depth ranges comprise the elements q_(j) ateach location j. One or more embodiments of the invention disclosedherein can predict the sonic log in the ranges 3043-3200 and 3250-3439and then apply that prediction it to predict the values in the range3201-3249.

In a preferred embodiment, three pre-processing steps are used. Thesethree steps are (1) log values are normalized to the range [−1, +1]; (2)logs that best predict the target log are selected; and (3) samples arechosen from these logs to predict the missing data.

Normalization compensates for the different units of log values. A gammaray log has values on the order of 100, while a resistivity log hasvalues on the order of 1. In the absence of normalization, traditionalnumerical analysis would give greater weight to the gamma ray log.

In a preferred embodiment, normalization is a procedure wherein valuesare offset and scaled so that the original range matches the givennormalization range. If the minimum value in a given log were zero andthe maximum 150, normalization would be accomplished by subtracting 75(the mid-point between zero and 150) and then dividing the result by 75(half the range). If the minimum were −40 instead of zero, thensubtracting 55 and dividing by 95 would normalize the log. The smallestvalue would be −1 and the largest would be +1.

Let L₀ designate the original set of logs excluding log m. In apreferred embodiment, we first omit logs with too many null values. Inanother preferred embodiment, we eliminate logs with more than 50percent null values. In other embodiments, a different null value cutoffmight be used. The set of logs L₁ represents those remaining from L₀after the elimination of logs with too many null values.

In a preferred embodiment, the next step is to cross correlate the logsin L₁. Logs in this set may be grouped according to similarity. In apreferred embodiment, only one log from each group is kept. This set oflogs, one from each group, is designated L₂.

In a preferred embodiment, the standard statistical correlationcoefficient is used for cross correlation. If the value of sample j inlog S is represented by S_(j), then the standard deviation is computedby

${\sigma_{S} = \sqrt{\left\{ {\left\lbrack {{n{\sum{Sj}^{2}}} - \left( {\sum{Sj}} \right)^{2}} \right\rbrack/n^{2}} \right\}}},$

where n is the number of samples. The same formula is used to calculateσ_(T) for log T. Then the average cross product is computed as

${C = {{\sum\limits_{\cdot}}_{{j = 1},n}\; \left\lbrack {{{Sj}*{{Tj}/n}} - {S_{avg}*T_{avg}}} \right\rbrack}},$

where S_(avg) and T_(avg) are average values for the logs. Thecorrelation, r, is

r=C/(σ_(S)*σ_(T)).

If the curves are identical, the value of r will be +1. If one is themirror image of the other, r will be −1. Logs with correlations neareither −1 or +1 are redundant and therefore too similar for both to bekept.

To decide which logs to keep, we group the logs by similarity and chooseone log from each group. Each log in a group is correlated with log m inthe zone ˜Z. The log with the highest correlation, r, is selected torepresent the group. Correlation is a measure of linear prediction, sowe are selecting logs that best predict the target log in the intervals˜Z using a linear equation. If there are n groups of logs, then the setL₂ will contain n logs.

We next find the subset of L₂ that best predicts m in the interval ˜Zusing the General Regression Neural Network (“GRNN”) algorithm. This isa test of non-linear correlation. These logs will be the set L₃. Onesuitable GRNN for use in predicting m in the interval ˜Z is described inSpecht, D. F. 1991, “A general regression neural network IEEETransaction on Neural Networks” at pages 568-576.

If there are n logs in L₂, n tests will be conducted. Logs are orderedby the linear correlation coefficient, and for each test the log withthe smallest correlation is dropped.

For each test, a portion of the samples are selected. Samples areselected by randomly choosing a portion of the ˜Z sampling depths commonto all logs. For example, if the common interval contains 120 samples,less than 120 will be selected. If a sample is null in any of the logsat the given depth, another depth is chosen, and the process is repeateduntil a desired number have been selected.

A first portion of these samples are designated, randomly, for trainingthe GRNN algorithm, and a second portion are designated for testing. TheGRNN algorithm uses each of the n sets of logs to predict values in logm, and each calculation produces a measure of how good the predictionis. The equation for scoring each set of logs is

S=w ₁ α+w ₂κ

where w₁ and w₂ are the scoring weights (we use w₁=1 and w₂=1).

α and κ measure accuracy and error. α is the average difference betweenthe predicted values and the actual values, divided by the maximumdifference−and subtracted from 1. In equation form:

α=1−Average |predicted−actual|/(maximum actual−minimum actual)

κ=correlation determined by the GRNN algorithm

If predictions are good, α and κ will be near 1, and S will be 2. Ifpredictions are poor, S will be nearer 0.

The set of logs with the largest score will be the set L₃.

We now predict values in the gap. Samples are selected that are similarto the samples at the given depth at which the log value is to bepredicted. FIG. 3 shows part of a table of numbers that could be usedfor this purpose. Values in the DT column are missing, with one row ofnumbers designated “Predictive set” and at depth of 3201. The objectiveis to use the eight numbers to the left to predict the missing value.

This is done by comparing this row of eight numbers, in interval Z, withall other rows in interval ˜Z. Similar rows have values in the DT columnand so form a basis for predicting the value at depth 3201. Excludingcolumn DT, we call the numbers in the interval Z set z and those in theinterval ˜Z, set ˜z.

Similar rows are found by using each number as a coordinate in ann-dimensional space. Points close together in this space are similar.The method is illustrated in FIGS. 4, 5, and 6.

FIG. 4 assumes there are only two columns of data, GR1 and DT. Rows inthe table at the right of FIG. 4 have been sorted by GR1. This is theorder in which they would appear if plotted on a line, as illustratedbelow the table. Blue points represent rows that have DT data, and redpoints represent rows that do not have DT data. Rows with numbers near−0.0976 would be chosen to make the estimate for depth 3201. The nearerthe values of GR1 were to −0.0976, the more weight would be given theirpredictions. When more than one column is used as a predictor, the rowscannot be posted as points on a line but must be posted in a higherdimensional surface.

FIG. 5 illustrates the analysis with three columns of data. A simplesort would not be effective, but a cross plot makes it clear whichtraining rows are “near” the target rows. We use the Pythagorean Theoremto calculate distance, but non-Euclidean measures might also be used.

FIG. 6 illustrates the plot when four columns are present. Points aredisplayed as if they were floating in space. The Pythagorean Theoremstill serves to compute the distance between points, and it can beapplied to as many dimensions as needed.

At this stage, Principle Components Analysis (“PCA”) may be applied toreduce the number of dimensions. This is particularly useful if thereare many logs. PCA makes use of correlations between measurements. Onesuitable method of PCA for use in reducing the number of dimensions isdescribed in Jolliffe, I. T., 2002, Principal Component Analysis,Series: Springer Series in Statistics, 2nd ed., Springer, N.Y.). FIG. 7illustrates the PCA method in two dimensions. The trend of points allowsus to replace the two original dimensions with one dimension. For eachsample, only the distance along the line (indicated by the arrowspointing from points toward the line) is used. Rather than using two logmeasurements only one measurement, along the principle axis, isrequired. PCA usually reduces the number of independent measures to twoor three.

The trend, or correlation, shows that information about log A allows oneto predict log B. The line drawn through the points, the line alongwhich measurements are made, eliminates the redundancy in theinformation.

FIG. 6 could also illustrate the analysis after PCA has been applied. Inthis case, measurements are along the principle axes rather than directlog measurements. For each sample in z (each red point), we compute itsdistance to each point in ˜z. For the sample z_(d) at depth d, thedistance to sample ˜z_(D) at depth D is

$D_{dD} = \sqrt{{Sum}\mspace{14mu} {of}\mspace{14mu} \begin{pmatrix}{{\log \mspace{14mu} {values}\mspace{14mu} {at}} \sim {z_{D} -}} \\{{corresponding}\mspace{14mu} \log \mspace{14mu} {values}\mspace{14mu} {at}\mspace{14mu} z_{d}}\end{pmatrix}^{2}}$

If D_(dD) is too large (we use a distance of D=0.5; other distances maybe used), the point in ˜z_(D) (a blue point) is dropped.

The GRNN algorithm is applied to the points within the required distanceto develop a predictor for the missing log. The predictor is thenapplied for each blue point and the predictions are averaged and theaverage is the value predicted for the red point.

We now compute the confidence, C_(d), in our prediction at depth d inset z. The total distance from z_(d) to all samples is

D _(d)=√{square root over (Σ_(D)(D _(dD))²)}

Our confidence in the prediction at this well is expressed by summingthe distances for all samples in z and applying a sigmoid function:

C _(d)=Sigmoid (Σ_(z)(1−D _(d)))

The sigmoid we use is a two-parameter logistics function:

Sigmoid (t)=1/(1+e ^(−k(t+t 0)))

The parameter k gives the slope of the sigmoid, and t0 is the offset. Wetypically use k=2 to 6 and t0=−2 to −6. FIG. 8 shows a typical sigmoid.

For each value in z we now have a predicted value and a measure ofconfidence in the prediction. The method for prediction within a singlewell is complete.

The analysis can be extended to several wells. Wells are analyzed one byone, and each well is treated in the same way as outlined here exceptthat 1) the randomly chosen test samples are always in the target wellwhile the randomly chosen training samples are in the other well and 2)there is no interval in the other well that corresponds to Z. Inaddition, if the score for the final set of logs, L₃ (S=w₁α+w₂κ) is toosmall, the well is not included in the analysis.

After the analysis has been performed on W wells with w wells rejected,we have N=W−w predictions and N confidence measures over the interval zin well log m. These can be thought of as N prediction curves and Nconfidence curves. The prediction curves are averaged to produce thefinal prediction, with the confidence of each value used as a weight inthe averaging.

Let Pd_(i) be the predicted values and Cd_(i) the confidence values atdepth d. The final prediction at depth d is

Pavg_(d)=Σ_(i=1,N) [Pd _(i) *Cd _(i)]/Σ_(i=1,N) [Cd _(i)]

Associated with each Pavg_(d) are the values, Pd_(i), that went into theaverage. These are used to measure the confidence in the finalprediction by casting them into the form of a density function.

Cfinal_(d) =KΣ _(i=1,N)exp{[(Pd _(i) −Pavg_(d))/σ]²}

where K is an arbitrary scaling factor, and σ is the standard deviationof Pd_(i).

This is perhaps best understood if the individual curves represented byexp {[(Pd_(i)−Pavg_(d))/σ]²}are plotted, as shown in FIG. 9. Each of thesmaller curves represents the function exp{[(x−Pd_(i))/σ]²} or itsequivalent exp{[(Pd_(i)−x)/σ]²}.

FIG. 9 shows curves resulting from three values of Pd_(i): Pd₁, Pd₂, andPd₃. The sum of these curves is the confidence value, and the value ofthe sum at Pavg_(d) is the confidence of this estimate.

We now have the final inter-well prediction for all values in the gap z,together with a confidence curve for the predictions.

1. A method for selecting samples of geological data comprising: a.acquiring a data set comprising multiple data values indicative of atleast one geological characteristic at locations in a subsurfaceinterval of rock; identifying locations i where (i E ∈ {1,n}) in asubsurface interval of rock where values v_(i) are to be predicted;identifying data values associated with each location i such that eachdata value associated with a location i is an element p_(i) and allelements p_(i) form set P; d. identifying locations j where (j ∈ {1,m})in a subsurface interval of rock where data values v_(j) from the dataset, are known; e. defining elements q_(j), to be samples v_(j) and dataat the same location as samples v_(j) at a multiplicity of locations jto form set Q; f. for an element p_(i) in set P, identifying one or moreelements q_(j) in set Q with data values indicative of the samegeological characteristic as the p_(i) sample is indicative of, suchthat the identified elements q_(i) constitutes data elements r_(i); andg. defining a data set R, comprising data elements r_(i).
 2. The methodof claim 1, where locations i and j are depth locations in a well. 3.The method of claim 2, wherein i comprises locations in a first well logcurve from a first well which is missing sample values.
 4. The method ofclaim 3, wherein p_(i) comprises sample values from a second well logcurve in the first well and at the same depths i.
 5. The method of claim1, wherein values v_(j) are known well log values.
 6. The method ofclaim 5, wherein the data at the same location as samples v_(j) at amultiplicity of locations j are log values at the same depth as v_(j).7. The method of claim 2, wherein j are locations in a first well logcurve from a first well which is missing sample values comprising datain wells other than the first well at depths where value v_(j) is known.8. The method of claim 1, further comprising associating an elementq_(j) with an element p_(i) by the Euclidean n-dimensional distance ofq_(j) from p_(i) using only elements q_(j) in set Q identified in step fof claim
 1. 9. A method for selecting samples of geological datacomprising: a. acquiring a data set comprising multiple data valuesindicative of at least one geological characteristic at locations in asubsurface interval of rock; b. identifying well depth locations i where(i ∈ {1,n}) in a subsurface interval of rock where values v_(i) are tobe predicted; c. identifying data values associated with each location isuch that each data value associated with a location i is an elementp_(i) and all elements p_(i) form set P; d. identifying well depthlocations j where (j ∈ {1,m}) in a subsurface interval of rock wheredata values v_(j) from the data set, are known; e. defining elementsq_(j), to be samples v_(j) and data at the same location as samplesv_(j) at a multiplicity of locations j to form set Q; f. for an elementp_(i) in set P, identifying one or more elements q_(j) in set Q withdata values indicative of the same geological characteristic as thep_(i) sample is indicative of, such that the identified elements q_(i)constitutes data elements r_(i); and g. defining a data set R,comprising data elements r_(i).
 10. The method of claim 9, wherein icomprises locations in a first well log curve from a first well which ismissing sample values.
 11. The method of claim 10, wherein p_(i)comprises sample values from a second well log curve in the first welland at the same depths i.
 12. The method of claim 9, wherein valuesv_(j) are known well log values.
 13. The method of claim 9, furthercomprising associating an element q_(j) with an element p_(i) by theEuclidean n-dimensional distance of q_(j) from p_(i) using only elementsq_(j) in set Q identified in step f of claim
 9. 14. A method forselecting samples of geological data comprising: a. acquiring a data setcomprising multiple well log data values indicative of at least onegeological characteristic at locations in a subsurface interval of rock;b. identifying locations i where (i ∈ {1,n}) in a subsurface interval ofrock where well log values v_(i) are to be predicted; c. identifyingwell log data values associated with each location i such that each datavalue associated with a location i is an element p_(i) and all elementsp_(i) form set P; d. identifying locations j where (j ∈ {1,m}) in asubsurface interval of rock where data values v_(j) from the data set,are known; e. defining elements q_(j), to be samples v_(j) and data atthe same location as samples v_(j) at a multiplicity of locations j toform set Q; f. for an element p_(i) in set P, identifying one or moreelements q_(j) in set Q with data values indicative of the samegeological characteristic as the p_(i) sample is indicative of, suchthat the identified elements q_(i) constitutes data elements r_(i); andg. defining a data set R, comprising data elements r_(i).
 15. The methodof claim 14, where locations i and j are depth locations in a well. 16.The method of claim 15, wherein i comprises locations in a first welllog curve from a first well which is missing sample values.
 17. Themethod of claim 16, wherein p_(i) comprises sample values from a secondwell log curve in the first well and at the same depths i.
 18. Themethod of claim 14, wherein the data at the same location as samplesv_(j) at a multiplicity of locations j are log values at the same depthas v_(j).
 19. The method of claim 15, wherein j are locations in a firstwell log curve from a first well which is missing sample valuescomprising data in wells other than the first well at depths where valuev_(j) is known.